Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623326 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,26)(13,16)(14,32)(15,35)(17,24)(18,27)(19,34)(20,31)(22,48)(23,45)(25,54)(28,60)(29,42)(30,46)(33,44)(36,43)(37,52)(38,53)(39,66)(40,64)(41,68)(47,49)(50,55)(51,69)(56,62)(57,59)(58,65)(61,67)(63,71)(70,72);(1,3,4,8)(2,5,6,7)(10,16)(17,37)(18,26)(19,38)(20,35)(21,45)(32,46)(39,55)(40,44)(41,56)(42,57)(43,54)(47,48)(60,65)(67,70)(69,71);(3,8)(5,7)(9,11,13,23)(10,17,16,37)(12,22,27,49)(14,33,30,64)(15,36,31,25)(18,39,26,55)(19,43,38,54)(20,44,35,40)(21,46,45,32)(24,53,52,34)(28,61,58,72)(29,63,59,51)(41,69,56,71)(42,70,57,67)(47,60,48,65)(50,68,66,62);(2,6)(5,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,29)(16,26)(17,39)(19,41)(20,42)(21,47)(23,49)(24,50)(25,51)(30,58)(31,59)(32,65)(33,61)(34,62)(35,57)(36,63)(37,55)(38,56)(40,67)(43,69)(44,70)(45,48)(46,60)(52,66)(53,68)(54,71)(64,72);(12,27)(18,26)(22,49)(28,58)(29,59)(39,55)(41,56)(42,57)(47,48)(50,66)(51,63)(60,65)(61,72)(62,68)(67,70)(69,71)Polycyclic presentation:
2^9|b^2=fi,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cfi,g^c=gh,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530579);